Numerical Approximation of Solutions to Stochastic Partial Differential Equations and Their Moments

The first part of this thesis focusses on the numerical approximation of the first two moments of solutions to parabolic stochastic partial differential equations (SPDEs) with additive or multiplicative noise. More precisely, in Paper I an earlier result (A. Lang, S. Larsson, and Ch. Schwab, Covariance structure of parabolic stochastic partial differential equations, Stoch. PDE: Anal. Comp., 1(2013), pp. 351–364), which shows that the second moment of the solution to a parabolic SPDE driven by additive Wiener noise solves a well-posed deterministic space-time variational problem, is extended to the class of SPDEs with multiplicative L\ue9vy noise. In contrast to the additive case, this variational formulation is not posed on Hilbert tensor product spaces as trial–test spaces, but on projective–injective tensor product spaces, i.e., on non-reflexive Banach spaces. Well-posedness of this variational problem is derived for the case when the multiplicative noise term is sufficiently small. This result is improved in Paper II by disposing of the smallness assumption. Furthermore, the deterministic equations in variational form are used to derive numerical methods for approximating the first and the second moment of solutions to stochastic ordinary and partial differential equations without Monte Carlo sampling. Petrov–Galerkin discretizations are proposed and their stability and convergence are analyzed. In the second part the numerical solution of fractional order elliptic SPDEs with spatial white noise is considered. Such equations are particularly interesting for applications in statistics, as they can be used to approximate Gaussian Mat\ue9rn fields. Specifically, in Paper III a numerical scheme is proposed, which is based on a finite element discretization in space and a quadrature for an integral representation of the fractional inverse involving only non-fractional inverses. For the resulting approximation, an explicit rate of convergence to the true solution in the strong mean-square sense is derived. Subsequently, in Paper IV weak convergence of this approximation is established. Finally, in Paper V a similar method, which exploits a rational approximation of the fractional power operator instead of the quadrature, is introduced and its performance with respect to accuracy and computing time is compared to the quadrature approach from Paper III and to existing methods for inference in spatial statistics

Numerical stability analysis of the Euler scheme for BSDEs

In this paper, we study the qualitative behaviour of approximation schemes for Backward Stochastic Differential Equations (BSDEs) by introducing a new notion of numerical stability. For the Euler scheme, we provide sufficient conditions in the one-dimensional and multidimensional case to guarantee the numerical stability. We then perform a classical Von Neumann stability analysis in the case of a linear driver $f$ and exhibit necessary conditions to get stability in this case. Finally, we illustrate our results with numerical applications

Discrete Razumikhin-type technique and stability of the Euler-Maruyama method to stochastic functional differential equations

A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations

Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations

A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

In this article we compare the mean-square stability properties of the Theta-Maruyama and Theta-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the Theta-Milstein method and thus, for some choices of Theta, the conditions on the step-size, are much more restrictive than those for the Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partially implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter Sigma. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.Comment: 19 pages, 10 figure

Time-Symmetric ADI and Causal Reconnection: Stable Numerical Techniques for Hyperbolic Systems on Moving Grids

Moving grids are of interest in the numerical solution of hydrodynamical problems and in numerical relativity. We show that conventional integration methods for the simple wave equation in one and more than one dimension exhibit a number of instabilities on moving grids. We introduce two techniques, which we call causal reconnection and time-symmetric ADI, which together allow integration of the wave equation with absolute local stability in any number of dimensions on grids that may move very much faster than the wave speed and that can even accelerate. These methods allow very long time-steps, are fully second-order accurate, and offer the computational efficiency of operator-splitting.Comment: 45 pages, 19 figures. Published in 1994 but not previously available in the electronic archive

Optimization of Nordsieck's Method for the Numerical Integration of Ordinary Differential Equations

Stability and accuracy of Nordsieck's integration method can be improved by choosing the zero-positions of the extraneous roots of the characteristic equation in a suitable way. Optimum zero-positions have been found by minimizing the lower bound of the interval of absolute stability and the coefficient of the truncation error. Various properties of the improved methods have been analysed, such as the behaviour of the high-order terms, the equivalence with multistep methods and the damping of perturbations